Remark 2.2 In [1]}, [2]}, Lusztig gave a categorification of the negative part \(\mathbf {U}^-\)  of a quantum group by using an additive subcategory \(\mathcal {Q}_{\nu }\)  of \({\mathcal {D}}_{G_{\mathbf {V}}}^{b,ss}(E_{\mathbf {V}})\) .
The multiplication on \(\mathbf {U}^-\)  is induced by induction functors \(\mathrm {Ind}^{\alpha +\beta }_{\alpha ,\beta }\)  and
the algebra \(\mathbf {K}\)  contains \(\mathbf {U}^-\)  as a subalgebra.

